Jamming-Aided Secure Communication in Massive MIMO
arXiv:1507.06521v1 [cs.IT] 23 Jul 2015 okhpo asv IO rmter opractice to theory From RCS20 MIMO: (No. 2014. Massive Safety and on Control Workshop Traffic 3 Rail (No. of Lab Education (2 Key of Universities State Central Ministry the the Chinese for Funds of 6120120 Research Project 61401240, Fundamental Grant Foundation Key Science the Natural National the n aey ejn ioogUiest,Biig104,C 100044, Beijing University, Jiaotong Beijing [email protected] Safety, and atn nvriy atn 209 China. 226019, Nantong University, Nantong EETASCIN NWRLS COMMUNICATIONS WIRELESS ON TRANSACTIONS IEEE oeUiest fTcnlg n ein igpr 487372 Singapore Design, and Technology of { University pore pu for it approving and paper this Han. of review the coordinating IOlast hr empten swl slwpower low as well as patterns massive beam beamforming, sharp and for to spectrum used leads in When MIMO growth the [4]. transmit/recei intensive efficiency infinity, linear to power to leading simple [3], goes only techniques antennas using to tend by of can noises asymptotically number years and interferences recent the uncorrelated in of the area effect As research at [2]. hot deployed a are [1], become antennas have of station, number base enormous an where effectiv schemes. the allocation show power jamming and proposed conclusions our our of last validate At exte scenarios. results further multi-cell ical We and eavesdroppers. multiuser to of suspicious discussions the locations of proposed information possible are the i.e., without algorithms and with allocation that cases power phys the jamming to and and directional (mapped only, beams efficiently propose selected angles) more at we signals power SOP, jamming generates the jamming reduce the further use to different betw for investigated allocation Furthermore, is power noise th optimal artificial reduce and the messages to Then, jamming determined. uniform is distance SOP, enables the which i.e., eavesdroppers range, of jamming-beneficial a probability outage and secrecy derived, the c that, on that After eavesdropper outage. an depends ( we secrecy of region locations light, outage channel possible secrecy this all Rician the indicating In describe a eavesdroppers. analytically that and of in define out locations happens wheth geometric figure infinity, outage largethe to We tends secrecy with channels. antennas transmit the equipped fading of number transmitter Rician the when a in for array design antenna jamming aided jue ato hswr a enpeetdin presented been has work this of Part F Research Temasek the by supported partly was Lee. research J. 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MIMO approaches, o emergence massive lately the secrecy, the most communication the the Regarding of [19]. F review jamming. precodi complete (AN)-aided noise a including artificial issues and characterizat scheduling, design region user transmission rate and practical analysis to s capacity contributions spa secrecy information-theoretical topics as from research range The wide [17]. channel a [16], MIMO [11], networks [10], and Corresponding wire channels [12]–[15], of fading [7]. type [9], different work [8], to seminal channels extended further the been have since studies decades for ied related security in crucial become also applications. anticipated will be can MIMO it generatio meanwhile, massive the fifth In the technique [6]. as system promising such cellular attracti systems a communication these future becomes to for MIMO Due massive [5]. directions properties, unintended to leakage nti ae,w td h euecmuiaini massive in communication secure the study we paper, this In stud- been has channels wiretap in communication Secure ebr IEEE, Member, eirMme,IEEE Member, Senior using that In . uch een ion ng, tap gle eir ve or re y, o y n g n e e s s 1 - f 2 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS and 2) as the number of antennas grows, the dimension of the jamming space increases and jamming power needs to spread over a large number of directions, which makes conventional uniform jamming inefficient with massive MIMO. Regarding these issues, two questions are raised: 1) Does conventional uniform jamming still benefit the secure communication in massive MIMO systems when Nt goes to infinity? 2) Is there more efficient scheme rather than uniform jamming in the massive MIMO setup? In this paper, we will answer these two questions by making the following contributions: • For the massive MIMO Rician fading channels, we analytically describe the secrecy outage region (SOR) as geometric locations of eavesdroppers that can induce Fig. 1: Description of the network layout. secrecy outage. The concept of SOR further has been used to characterize the secrecy outage probability (SOP). • With the information of the suspicious area where eaves- droppers are possibly located, we derive analytical ex- with AN in other spaces (or directions). We define the set pression of the SOP in the presence of one legitimate of receivers Ir = {b, e1, ..., eL} where b denotes Bob and el receiver and multiple passive eavesdroppers. After that, (l = 1, ..., L) denotes Eve l. Considering Rician fading, the it is proved that conventional uniform jamming is still channel between Alice and receiver i is given by useful in terms of reducing the SOP when any eaves- Ki 1 droppers are located within a certain distance range to hi = h¯i + gi, ∀i ∈ I (1) 1+ K 1+ K r Alice, which we call it as the jamming-beneficial range. r i r i N ×1 This conclusion provides an answer to the first question. where Ki is the Rician K-factor, gi ∈ C t is the i.i.d. • For uniform jamming, the optimal signal and jamming fast fading part whose elements follow CN (0, 1) distribution power allocation is investigated for different scenarios. (complex normal distribution with zero mean and unit vari- We further devise practical directional jamming algo- ance). For uniform linear array with inter-antenna spacing d0 rithms, either with or without the information of the sus- (in wavelength), the line of sight (LOS) component h¯i can be picious area. The proposed directional jamming schemes written as the steering vector at incident angle θi: use the jamming power more efficiently to further and T ¯ −j2πd0 sin θi −j2π(Nt−1)d0 sin θi substantially reduce the SOP, which provides answers to hi = ¯s(θi)= 1,e , ..., e the second raised question. (2) The rest of this paper is organized as follows: Section II where θi is the LOS angle of receiver i. In addition, we −α provides system model. Section III describes the SOR, further consider large scale fading di where di is the distance from provides an analytical expression of SOP and a jamming- Alice to receiver i, and α is the path loss coefficient. beneficial range. Optimal jamming power allocation is stud- We consider a practical scenario that Eves are uniformly , ied for uniform jamming in Section IV, and in Section V, distributed within an angular range Ae [θmin,θmax] and a , directional jamming algorithms are proposed. In Section VI, distance range De [Dmin(θe), Dmax(θe)], where Dmin(θe) the SOR is discussed for multiuser and multi-cell scenarios. and Dmax(θe) are functions of θe ∈ Ae, defining two borders Section VII concludes this paper. of this area. Throughout this paper, we use
Rsus , {(θe, de) | θe ∈ Ae, de ∈ De} (3) II. SYSTEM MODEL to define the suspicious area. In practice, if Alice has only lim- In this section, we first present the network model. As an ited information of D (θ ) and D (θ ), she can assume important concept in subsequent analysis, we further define the min e max e the two boundaries are defined by constant values, d and normalized crosstalk between two wireless links and introduce min d . For instance, if Alice knows nothing about R , she can its characteristics. Then, the AN-aided secure transmission and max sus set A = [0, 2π], D = [0, r ], indicating that the suspicious the definition of SOP are described. e e max area (from Alice’s point of view) spans the entire space with radius rmax. The effectiveness of this assumption, referred to A. Network Model as “constant boundaries” and defined below, depends on that We consider the network shown in Fig. 1, where a transmit- how accurately it can describe the real Rsus. ter (Alice) equipped with Nt antennas transmits to a single- Definition 1 (Constant Boundaries): To facilitate practical antenna user (Bob) in the existence of L external passive design, it is convenient to set the two boundaries of Rsus single-antenna eavesdroppers (Eves 1, ..., L). Alice uses beam- to be constants such that Dmin(θe) = dmin, Dmax(θe) = forming for the data transmission to Bob, while jamming dmax, ∀θe ∈ Ae. 3
1 − F CP ( x ) , PV ≤ x ≤ 1 ∆ 0 Ki;j 1 Ki;j Fs (x)= M (7) i;j x x x x 1 − F CP ( ) + F CPm, ( ) − F CPm, ( ) , PVM ≤ < PVM ∆ 0 Ki;j ∆ 1 Ki;j ∆ 2 Ki;j +1 Ki;j m=1 P
B. Normalized Crosstalk allocation coefficient as N For hi, hj (j ∈ I ) defined as (1), the following asymptotic P P¯n r φ , jam = n=1 . (9) results hold as Nt → ∞ [30]: P P tot P tot 1 H . For ease of description, we assume that Bob and all Eves hi hi =1 (4) Nt share the same noise covariance being N0. Moreover, we 1 H . 1 consider maximum ratio transmission (MRT) for precoding of h hj = Ki jti j , j 6= i. (5) i ; ; hb Nt Nt the data symbol xb, i.e., wb = ||h || . In this case, according . b where = denotes the approximationp that is asymptot- to (8), the SINR at receiver i is given by 1 KiKj α ically accurate, Ki;j , , and ti;j , − H 2 (1+Ki)(1+Kj ) Pbdi |hi wb| Nt−1 j πd θ θ n SINRi = . (10) e− 2 0(sin i−sin j ) . Stemming from (5), we intro- −α N ¯ H 2 n=0 N0 + di n=1 Pn|hi vn| duce the following definition. We assume that the Eves are not colluding, but consider the PDefinition 2 (Normalized Crosstalk): Define the normal- P most-capable Eve, which has the maximum receive SINR, to ized crosstalk between nodes i, j as define the secrecy rate as [22] 2 , 1 H . 1 2 s + si;j (θi,θj ) hi hj = 2 Ki;j |ti;j | . (6) Rb = [log2(1 + SINRb) − log2(1 + SINRe,max)] (11) Nt Nt , + , where SINRe,max max SINRel and [x] max{x, 0}. We Lemma 1: The normalized crosstalk si;j (θi,θj ) has the l s following characteristics: say a secrecy outage occurs if Rb is less than a target rate Rth, hence the SOP is defined as 1) si;j (θi,θj ) is a sinc-like function composed of one main s lobe and multiple side lobes. Pout = Pr{Rb < Rth}. (12) 2) With fixed θj and random θi ∼U(θmin,θmax) which is uniformly distributed between θmin and θmax, the CDF III. SECRECY OUTAGE ANALYSIS of si;j (θi,θj ) can be written as (7) (see top of this page) where the definitions of CP0, CPm,1, CPm,2 and F∆(·) In this section, we first introduce the secrecy outage region are referred to Appendix A. (SOR) which describes all possible locations of Eves who can 3) With fixed θj , the feasible range of si;j (θi,θj ) is 0 ≤ cause secrecy outage. Analytical expression of the SOR is max max si;j ≤ si;j , where si;j ≤ Ki;j is determined by the derived for uniform jamming, then the SOP is studied with distribution range of , i.e., , . θi Ai [θmin,θmax] variant shapes of Rsus. At last, a jamming-beneficial range Proof: See Appendix A. is derived to show that uniform jamming is still useful in reducing the SOP. C. Secrecy Transmission Scheme We use linear precoding for data transmission, while AN A. Secrecy Outage Region symbols sn are sent in the space defined by vn,n =1, ..., N, In the large antenna regime, all fast fading effects are to degrade the channels of Eves. For the null space-based completely averaged out as shown in (4) and (5). Therefore, jamming [27], it holds that N = Nt − 1 and vn ∈ null(hb). whether the secrecy outage occurs or not, will be essentially The received signal at receiver i is given by determined by the geometric location of Eve. In this light, we N introduce the SOR defined in the following. −α H ¯ −α H yi = Pbdi hi wbxb+ Pndi hi vnsn+ni, ∀i ∈ R Definition 3 (Secrecy Outage Region): The SOR is defined n=1 in terms of polar coordinates as q X q (8) CNt×1 where wb ∈ is the precoder for Bob, xb is the unit- , s RSOR (θe, de) | lim Rb < Rth . (13) norm data symbol, and n is the additive Gaussian noise. Nt→∞ i Moreover, Pb and P¯n respectively are the powers allocated s Herein, we note that R is a function of θe and de. to Bob and the -th jamming direction, with total power b n In the large antenna regime, secrecy outage occurs if there constraint such as N ¯ where is the n=1 Pn = Ptot − Pb Ptot exists at least one Eve within the SOR. If all Eves locate total available transmit power. We define the jamming power P outside of the SOR, the target secrecy rate Rth can be guaranteed. To characterize the SOR, we first evaluate the 1In this paper, we focus on the large antenna regime, and will use equalities instead of approximations for brevity. received SINRs assuming uniform jamming. 4 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS
−α Rth 1+(1 − φ)P˜totd Nt − 2 P˜totφ C2(φ) b C3(φ)= = . (22) α C1(φ) ˜ − Rth ˜ ˜ Rth 1+(1 − φ)Ptotdb Nt − 2 Ptotφ + (1 − φ)PtotNt2
Lemma 2: With uniform jamming in null(hb), the SINRs Proof: Substituting (14) and (15) into (11) and using the 2 ¯uj at Bob and any Eve can be, respectively, written as definition of SOR in (13), we can obtain the value of de (φ, θe) in (19). Note that the value of d¯uj(φ, θ ) should be positive, SINRuj = P˜ d−αN (14) e e b b b t this straightforwardly introduces the constraint on the mini- ˜ −α uj Pbde Ntse;b(θe) mum value of se;b(θe) such as se;b (θe) > C3(φ) where C3(φ) SINRe = −α (15) 1+ de P˜jam(1 − se;b(θe)) can be readily obtained by letting C1(φ)se;b(θe)−C2(φ) > 0.
Pb where, and hereafter, we use the notations P˜b = , P˜jam = N0 From Lemma 1, se;b(θe) in (19) is a function with one main Pjam for brevity. Note that in (15), s (θ ) is the normalized N0 e;b e lobe and multiple side lobes, resulting in a multi-lobe shaped crosstalk between Eve and Bob as defined in (6). Considering SOR. In order to gain some insights from this complex shape fixed θb, we hereafter write se;b as a function of only θe. (as will be shown in the simulations), we focus on several Proof: Since span(vn) = null(hb), jamming causes no critical security-related metrics as interference at Bob. Applying (4) to (10), we get (14). Noting ¯uj Pjam • Largest radius of the main lobe (de,0) and the m-th side that N = Nt − 1 and P¯n = , from (10), we have Nt−1 ¯uj ¯uj ¯uj lobe (de,m): de,0 and de,m can be obtained by replacing −α H 2 uj Pbde |he wb| se;b(θe) in (19) respectively with Ke;b and Ke;bPVm SINRe = . (16) Nt−1 (defined as (54) in the proof of Lemma 1), such that −α Pjam H 2 N + de |h wn| 0 Nt−1 e 1 n=1 ¯uj , α de,0 (C1(φ)Ke;b − C2(φ)) (23) P 1 By applying (6), the numerator of (16) can be written ¯uj , α −α de,m (C1(φ)Ke;bPVm − C2(φ)) . (24) as Pbde Ntse;b(θe). On the other hand, noting that wb ¯uj ¯uj ¯uj and vn,n = 1, ..., Nt − 1 constitute a complete orthog- Since de,0 is much larger than de,m, ∀m 6=0, de,0 can be onal basis of the Nt-dimensional vector space, we have considered as the largest distance of the SOR. For any Nt−1 2 2 2 ¯uj n=1 |hewn| = ||he|| − |hewb| = Nt(1−se;b(θe)) in the Eve whose distance to Alice is larger than de,0, we can denominator of (16). Therefore, (15) can be obtained. conclude that it causes no secrecy outage regardless of PRemark 1: The result in (14) leads to a constraint on φ its LOS direction. (defined in (9)), written as • Largest angle difference ∆θmax of the SOR: For any 2Rth − 1 Eve whose angle difference to the LOS direction of Bob φ ≤ φmax =1 − (17) ˜ −α is larger than ∆θmax, we can conclude that it causes Ptotdb Nt no secrecy outage regardless of its distance to Alice. If which stems from the fact that the jamming power cannot be se;b(θe) < C3(φ), ∀θe > θˆe, we can write too large, otherwise, even without Eves, the target rate R th ˆ cannot be guaranteed since the remained signaling power is ∆θmax = θe − θb . (25) too small. Unless otherwise specified, we assume (17) can Clearly, to have smaller SOR, we expect to reduce both ¯uj always hold via proper power allocation. de,0 ¯uj From Lemma 2, we characterize the SOR for the uniform and ∆θmax. To minimize de,0 in (23), we need to minimize jamming as follows. C1(φ) while maximizing C2(φ); however, since C3(φ) = C2(φ) , this results in a maximized C (φ) (correspondingly, Proposition 1: With uniform jamming in null(hb) and C1(φ) 3 given φ, the SOR is described as a larger ∆θmax), which is not desired. It is clear that a trade- ¯uj off in φ exists in balancing the effects of both de,0 and uj ¯uj ∆θmax, which can be formulated as jamming power allocation RSOR(φ)= (θe, de) | de < de (φ, θe),se;b (θe) > C3(φ) ( ) problems, as described in the following sections. (18) At last, by setting φ = 0 in Proposition 1, we obtain the where following corollary: uj 1 Corollary 1: Without jamming, i.e., φ = 0, the SOR can d¯ (φ, θ ) = (C (φ)s (θ ) − C (φ)) α (19) e e 1 e;b e 2 be found as ˜ Rth (1 − φ)PtotNt2 1 C (φ)= + P˜ φ, (20) α 1 −α tot Rth ˜ Rth P˜totNt2 se;b (θe) 1+(1 − φ)Ptotdb Nt − 2 nj RSOR = (θe, de) | de < −α ˜ Rth ˜ (21) 1+ Ptotdb Nt − 2 ! C2(φ)= Ptotφ (26) and is given by (22) shown at the top of this page. C3(φ) where the superscript (·)nj stands for “no jamming”. 2Hereafter, if one notation is applied for any Eve, we will use the subscript Proof: The corollary is directly obtained by setting φ =0 e instead of el for the sake of brevity. in Proposition 1. 5
s Differently from (18), the constraint on se;b(θe) vanishes in (12) as Pout =1 − Pr{Rb ≥ Rth} and recall (11), we have (26), indicating that the SOR now is extended to the entire uj angular domain. Moreover, compared with (23), we see that s 1 + SINRb Pr{Rb ≥ Rth} = F uj − 1 (31) ¯ 3 SINRe,max Rth de,0 in no-jamming case, on the contrary, is reduced compared 2 ! to uniform jamming. In conclusion, uniform jamming induces uj where F uj (·) is the CDF of SINR , which is two opposite effects: the beneficial one is that the SOR can SINRe,max e,max L be squeezed in angular domain, and the disadvantage is that given by F uj (x) = F uj (x) since all Eves are SINRe,max SINRe the SOR is enlarged in Bob’s direction, i.e., the main lobe. independently distributed. Using (15), we have Illustration of the SOR changing caused by jamming will be α ˜ shown later in simulations. de + Pjam F uj (x)=Pr se;b(θe) ≤ (32) SINRe P˜bNt ( + P˜jam ) B. SOP Analysis x where both θ and d are random. Since θ and d are With a single Eve uniformly distributed in , using the e e e e Rsus independent, (32) can be presented as derived SOR, the SOP is given by dmax α ˜ Area (R (φ) ∩ R ) z + Pjam SOR sus FSINR (x)= Fs fd (z)dz. (33) Pout,singleEve = (27) e e;b P˜ N e d b t ˜ Area (Rsus) Z min x + Pjam ! where denotes the area of a certain geometric region. , 2z Area(·) where fde (z) 2 2 is the PDF of de, corresponding to dmax−dmin Considering that there are L Eves uniformly distributed in the uniform distribution between two boundaries defined by , the SOP of the entire network can be written as Rsus De = [dmin, dmax]. Then, Pout is directly obtained as (30). L Practically, (30) can be used for jamming power alloca- P =1 − (1 −P , ) . (28) out out singleEve tion. As stated in Remark 1, Alice can arbitrarily adjust the From (27), the SOP is determined by the overlapping area value of the constant boundaries in the design, based on the between two geometrical regions. If RSOR(φ)∩Rsus = ∅, zero information about Rsus that she has. Particularly, if Alice SOP is achieved. Recalling (3), as well as (23) and (25), two knows nothing about Rsus (i.e., she assumes Ae = [0, 2π] sufficient conditions of RSOR(φ) ∩ Rsus = ∅ can be written and De = [0, rmax]), minimizing Pout becomes equivalent to as minimizing Area (RSOR(φ)). ¯ Dmin(θe) > de,0, ∀θe, C. Jamming-beneficial Range or Dmax(θe)=0, ∀ |θe − θb| < ∆θmax (29) Based on Proposition 2, we find a jamming-beneficial range defined in d (i.e., the larger constant distance boundary of where d¯e,0 is defined in (23) and ∆θmax is in (25). The max physical insight of (29) is clear: when an Eve is far away Rsus) as follows. Proposition 3: A constraint on d that makes the uniform or its angle difference to θb is large, it does not cause outage. max jamming beneficial in reducing the SOP is given by For the general case with arbitrary shape of Rsus, Pout 1 in (27) can be numerically evaluated and further applied to α ˜ Rth jamming power allocation design. However, due to the non- max PtotNt2 dmax < se;b (34) ˜ −α Rth regular shapes of RSOR(φ) and Rsus, closed-form expressions 1+ Ptotdb Nt − 2 ! of Area (RSOR(φ) ∩ Rsus) as well as the SOP in (28) do max where se;b is the largest feasible crosstalk value defined in not exist for the general case. Yet, by considering constant Lemma 1. boundaries of Rsus as described in Definition 1, (27) can be Proof: See Appendix B. written in an integral form as the following proposition. Remark 2: Proposition 3 shows that when Eves are located Proposition 2: With constant boundaries of Rsus, i.e., close enough to Alice, uniform jamming is always beneficial Ae = [θmin,θmax] and De = [dmin, dmax], and uniform in reducing the SOP. Clearly, this range expands with larger jamming in null(hb), the SOP can be given as max db, as well as larger se;b or larger Rth. On the other hand, the range shrinks with larger Nt or P˜tot. dmax α ˜ z + Pjam Moreover, we note that (34) has a similar form of that Pout =1 − Fse;b ˜ R nj PbNt2 th ˜ ( dmin −α R + Pjam described for the SOR without jamming, i.e., R in (26). Z 1+P˜ d N −2 th SOR b b t Recalling the definitions of the largest distance of SOR in (23) L 2z and (24), the physical insight of Proposition 3 can be explained × dz . (30) nj 2 2 as follows: as long as R ∩ R 6= ∅, there always dmax − dmin ) sus SOR exists an optimal φ, with which the SOP can be reduced by where Fse;b (·) is defined in (7). uniform jamming, compared with the SOP without jamming. Proof: For the ease of analytical description, herein we The optimization of φ is discussed in the next section. utilize the CDF of the normalized crosstalk in (7). First, rewrite IV. JAMMING POWER ALLOCATION 3Hereafter, for the conditions where the corresponding notation can be applied for either the uniform-jamming or no-jamming cases, we ignore the In this section, considering uniform jamming, we investigate superscript “uj” or “nj” for brevity. the optimal jamming power allocation that minimizes the SOP. 6 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS
The problem can be simply described as causing secrecy outage, hence should be treated differently in the jamming design. min Pout, s.t. 0 ≤ φ ≤ 1. (35) φ B. Jamming with Exact Information of R In practice, Alice may have different accuracy levels of sus With the information of Rsus, Alice can calculate and apply information about Rsus as follows: the value of P in the design (at least numerically),4 using 1) Alice knows nothing about the suspicious area, or only out either (28) or (30). Although in practice, (35) can be readily partial information about the suspicious area such as Ae solved by one dimensional linear search, it fails to provide only (or D only); and e the optimal φ in closed form. In the following corollary, we 2) Alice knows exact information about the suspicious area, provide closed-form solutions and discussions for a special i.e., both and . Ae De case. For these two cases, we respectively investigate the jamming Corollary 3 (Jamming power allocation for given θe): For power allocation in the following. constant boundaries of Rsus and given θe, which is equal for all Eves, the optimal φ can be determined as opt A. Jamming with None/Partial Information about Rsus opt φg , φ0 ∈/ [0, 1] φ = opt (39) min{φg , φ0} φ0 ∈ [0, 1] When Alice knows nothing about Rsus, minimizing the SOP becomes equivalent to minimizing the area of SOR, which can where R R be calculated as R (2 th −1)2 th se;b(θe) (2 th − 1)+ Nt 2π opt 1−se;b(θe) 1 2 φ =1 − , (40) Area Ruj (φ) = d¯uj(φ, θ ) dθ (36) g q−α ˜ SOR e e e db NtPtot 0 2 ˜ Rth Z Ptot Nt2 se;b(θe) α −α − d uj uj ˜ Rth min where ¯ is defined in (19). Note that is 1+Ptot db Nt−2 de (φ, θe) RSOR(φ) φ = . (41) uj 0 ˜ composed of many side lobes. We use RSOR,m(φ) to denote (1 − se;b(θe))Ptot uj the m-th side lobe, and RSOR,I(φ) to denote a group of Proof: See Appendix D. opt opt side lobes with indices described by the set I. For the case Note when φ = φg , the optimal jamming power that Alice knows Ae or De, we can simplify the problem by decreases with d and s (θ ), whereas it will increase when uj b e;b e minimizing partial, other than the entire area of such opt RSOR(φ) φ = φ0. The part that dominates the final result in (39) as depends on the value of θe. Detailed discussions will be uj uj provided in Section VII along with simulations. Area RSOR,I′ (φ) = Area RSOR,m(φ) (37) m ′ X∈I V. DIRECTIONAL JAMMING ′ where I is the set of the concerned side lobe indices, In this section, we propose directional jamming algorithms determined by either Ae or De. Using (36) (or (37)) along to allocate jamming power more efficiently than uniform with (19), the areas can be numerically calculated and the jamming, based on the following facts: optimal φ can be easily founded via one dimensional linear 1) With the information of Rsus, Alice can perform jam- search. Since it is difficult to derive closed-form expression ming only to the suspicious directions instead of the for uj , we evaluate the area of uj Area RSOR,I (φ) RSOR,m(φ) entire null space of hb. in the following corollary for a special case to further provide 2) Without information of Rsus, jamming towards different some discussions. directions also needs to be treated differently, as stated uj Corollary 2 (Area of RSOR,m(φ)): With θb = 0 and the in Remark 3. path loss coefficient being α = 2 (which corresponds the At a cost of slightly increasing the implementation complexity uj free space propagation [32]), Area RSOR,m can be upper compared with uniform jamming, directional jamming is able bounded as to substantially reduce the SOP. In following subsections, we present power allocation algorithms for directional jamming uj 1 Ke;b Area RSOR,m(φ) ≤ 2 2 C1(φ) − 2C2(φ) with and without the information of Rsus. 4Ntd π m (38) A. Directional Jamming with the Information of Rsus where C1(φ) and C2(φ) are defined in (20) and (21). Proof: See Appendix C. When jamming is not uniformly performed, from (10), the Remark 3: In (38), it is shown that the area of every side SINR at Eve is represented as lobe is inversely proportional to Nt, indicating that the SOR −α H 2 dj Pbde |he wb| side lobes can asymptotically vanish with ultimately large N . SINRe = α (42) t N + d− hH Vdiag (p¯) VH h Moreover, it is inversely proportional to m2, which means 0 e e e 4 that the area of the SOR will rapidly decrease for the side The calculation requires the knowledge of De and Ae. Clearly, uniform lobes with large indices, i.e., with large angle difference to jamming is not optimal in this condition. However, for the ease of analysis, we first devise the optimal power allocation for uniform jamming; then, the θb. This result indicates that Eves from different directions resulted jamming power can be allocated directionally to further improve (i.e., within different side lobes) have different significance in efficiency. 7
Algorithm 1 Directional jamming with the information of Algorithm 2 Iterative directional jamming without informa- Rsus tion of Rsus
1: Initialization: Update the information of θb, db and Rsus 1: Initialization: Update the information of θb and db at at Alice. Alice; set an initial jamming power allocation vector p¯0, max 2: Assuming null space-based uniform jamming, find satisfying ||p¯0|| ≤ Ptotφ ; set the initial iteration index opt j =1. φ = arg min Pout φ 2: for n =1 to N do 3: Update through one dimensional linear search over φ ∈ [0, 1). 3: Select a N-dimensional subspace (vn,n = 1, ..., N and dj N ≤ Nt) from null{hb} according to (46), where θvn is p¯j (n) = arg min Area RSOR [p¯j (1), ..., defined as (45). x∈[0,xmax) opt opt 4: Equally allocate Pjam = φ Ptot to the selected beams P opt p¯j (n − 1), x, p¯j−1(n + 1), ..., p¯j−1(N)] in step such as jam . 3 N ! n−1 N max where xmax = Ptotφ − p¯j (i) − p¯j−1(i). i=1 i=n+1 where V ∈ CNt×N is the matrix that spans the jamming space, 4: end for P P ¯ ¯ T with the n-th column vector being vn, and p¯ = P1, ..., PN , 5: if ||p¯j − p¯j−1|| ≥ ε then ¯ where Pn is the power allocated to the n-th jamming direction 6: j = j +1; go to step 2. as defined in (8). Correspondingly, the SOR now can be 7: else if ||p¯j − p¯j−1|| <ε then described as 8: return dj ¯dj 9: end if RSOR(p¯)= (de,θe) | de ≤ de (p¯,θe) (43) Rth ˜ ¯dj 2 (1 − φ)PtotNtse;b (θe) de (p¯,θe)= α ˜ − Rth "1+(1 − φ)Ptotdb Nt − 2 vn as the column vectors of a Nt-dimensional DFT matrix for 1 + the following reasons; 1) selected columns of the DFT matrix α H H − ¯s(θe) Vdiag(p˜)V ¯s(θe) (44) can form a good substitute of null(hb), as Nt → ∞ [33]; # 2) using pre-defined DFT basis as the jamming space avoids channel inverse calculation, which induces high computation where p˜ , p¯ , ¯s(·) was defined in (2). The superscript (·)dj N0 complexity especially when N is large [28]; and 3) most stands for “directional jamming”. t importantly, the structure of DFT matrix provides very sharp Design directional jamming using (42) induces high com- beam pattern towards the physical angle θ in (45), therefore, plexity especially when N is large, since changing any ele- vn the beam selection criterion (46) can be very efficient since ment in the N-dimensional vector p¯ requires re-calculation with sharper beams, there will be less jamming power leaked of Rdj (p¯). Hence, we alternatively propose a two-step SOR outside of R . suboptimal power allocation method for directional jamming sus in Algorithm 1, which firstly find the optimal jamming power assuming uniform jamming, then reallocate it directionally B. Directional Jamming without Information of Rsus based on a criterion of jamming subspace selection. In Step Without any information of R , the objective of direc- , can be calculated numerically using (28) or (30), sus 2 Pout tional jamming power allocation becomes to minimize the area depending on the available information of Rsus. Note that dj of RSOR(p¯) in (43) for θe ∈ [0, 2π], which is calculated as θb, db and Rsus are long term parameters, thus the updating 2π period of Step 1 can be much longer than the computation dj 1 ¯dj 2 Area RSOR(p¯) = de (p¯,θe) dθe. (47) time required by the other steps. 0 2 Since is defined by physical angles, it is necessary to Z Rsus A general closed-form expression of (47) is not available, set up a mapping between the jamming space and physical and its convexity is unknown. Hence, numerically minimizing angle to concentrate the jamming power towards R . We sus Area Rdj (p¯) is NP-hard. To overcome this, we propose propose a heuristic subspace selection method for Step 3. First, SOR Algorithm 2, which iteratively finds the optimal n-th element map vn to physical angle as of p¯ while keeping the others fixed. T 2 θvn =arg max ¯s(θ) vn . (45) Algorithm 2 provides a sub-optimal solution which reduces θ∈[− π , π ] 2 2 the complexity by degrading the original problem to one-
opt dimensional linear search. However, for large , the complex- After that, Pjam is equally reallocated to the beams whose Nt indices are ity is still huge since during each main iteration, N ∼ O(Nt) , times of linear searching are required to fully update p. Hence, N {n | θvn ∈ Ae}. (46) ¯ opt Algorithm 2 is not suitable for some scenarios where θb and db Pjam The power allocated to each beam is now dim(N ) . In practice, change fast. In this light, we propose a simplified algorithm, vn is not necessarily in null(hb) and an alternative is to find Algorithm 3, to further reduce the complexity. In Step 2 of 8 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS
Algorithm 3 Simplified directional jamming without informa- A. Multiuser Transmission tion of R sus When multiple legitimate users (i.e., multiple Bobs) are 1: Initialization: Update the information of θb and db at presented in massive MIMO systems for Rician channels, Alice; Initialize Areamin = Inf.; the multiuser interference between Bobs is trivial as long ′ ′ 2: Calculate se;b(θm),m =1, ..., M. Determine m1,m2, sat- as their LOS angles have large difference, which can be isfying that s (θ ′ ) ≥ s (θ ′ ) ≥ s (θ ), ∀m 6= m′ e;b m1 e;b m2 e;b m 1 readily ensured via user scheduling. On the contrary, the ′ and m 6= m2; multiuser interference to Eves can be seen as equivalent 3: for φ =0 to 1 do jamming considering that single-user decoder is adopted at 4: Find Eves, which is likely to happen when Eves are low-cost opt dj devices. Hence, when multiuser beamforming is applied for p¯ = argminArea R (p¯boundary) . boundary SOR Bobs, the received multiuser interference at Eve becomes equal dj opt to the directional jamming, transmitted towards other Bobs’ 5: Calculate Area p¯ according to (47); SOR boundary directions. Consequently, the SOR of an objective Bob will dj opt 6: if AreaSOR p¯boundary < Areamin then be shrunk in the directions of the other Bobs. Denote the set MU dj opt of all legitimate users as Ir = {b1, ..., bU }. Similar to (43) 7: Areamin = AreaSOR p¯boundary ; opt opt opt and (44), which describe the SOR for directional jamming, the 8: φ = φ; p¯ = p¯boundary . MU SOR of user bu ∈ Ir in the presence of multiple users can 9: else now be described as 10: continue MU,bu ¯MU,bu 11: end if RSOR (p¯bu )= (de,θe) | de ≤ de (p¯bu ,θe) (49) 12: end for R 2 th (1 − φ)P˜ Nts (θ ) d¯MU,bu (p¯ ,θ )= tot e;bu e e bu e −α R 1+(1 − φ)P˜totd Nt − 2 th " bu Algorithm 3, θm is the mean angle of the m-th side lobe of 1 + dj α R (p¯) (m =0 denotes the main lobe). In Step 4, p¯ H H SOR boundary − ¯s(θ ) Vdiag(p˜ )V ¯s(θ ) (50) follows the structure such as e bu e # p¯ = 0, ..., P¯ ′ , 0, ..., P¯ ′ , 0, ... , CNt×N boundary m1 m2 where V = [Wb, Vj] ∈ spans the equivalent jamming ′ ′ P¯ ′ + P¯ ′ = φP (48) space, in which Wb = wb ′ ,u = 1, ..., U, u 6= u m1 m2 tot u spans the signaling space for the other legitimate users, where m′ and m′ are the indices of the two dominating side 1 2 while Vj ∈ null (Wb) is the jamming space. Corre- lobes, which are located most closely to the main lobe (from spondingly, the power allocation vector can be divided into both sides). The derivation of Algorithm 3 is described in T T T two parts as p˜ = p˜ , p˜ where p˜ , , Appendix E. bu bu,sig bu,jam bu sig T With Algorithm 3, jamming is performed in only two dom- ˜ ˜ ˜ h ˜ i Pb1 , ..., Pbu−1 , Pbu+1 , ..., PbU is the signal power alloca- inating directions in the neighborhood of θ , for the reasons b tionh vector for all legitimate usersi except for bu. For given that 1) for the region with large angle difference to Bob, , fixed p˜ , in order to minimize Area RMU bu (p¯ ) for allocating much jamming power is inefficient since Rnj in bu,sig SOR bu SOR user , the directional jamming algorithms, i.e., Algorithm this region is generally very small; 2) for the directions highly bu 2 and 3, can be directly applied herein. in-line with θ , jamming should be avoided as it will cause b Considering communication secrecy for the entire multiuser severe interference to Bob. Note that for every realization of transmission system, the optimization problem can be reason- φ, only single time of linear search is required in Step 4. ably re-formulated as a min-max problem such as The complexity is irrelative to N (which is large in general), hence can be greatly reduced compared to Algorithm 2. As MU,bu min max Area R (¯pb ) ¯p SOR u a possible extension, more than two dominating directions bu bu (51) MU can be involved in the design while the trade-off between s.t. k¯pbu k1 < Ptot, bu ∈ Ir . complexity and performance exists. The main challenge in solving (51) is that allocating power for one user affects the SORs of other users. Hence, the power needs to be jointly allocated for all users, and the complexity of such joint optimization can be very high. Hence, it is VI. EXTENSION TO MULTIUSER AND MULTI-CELL desirable to develop simplified algorithms for the multiuser SCENARIOS scenario.
We focus on the single-cell and single-user scenario in previous sections. In this section, we now show how the B. Multi-cell Network SOR can be affected by multiple users and cells. We also In multi-cell massive MIMO networks, it is commonly provide discussions on the design of secure transmission in assumed that the training pilots are reused among cells. these scenarios with future research challenges. Correspondingly, pilot contamination results in imperfect CSI 9
1500 1
Nt = 100, Uniform Jamming 0.9 Nt = 100, Directional Jamming N = 50, Uniform Jamming 0.8 t (m)
0.7 uj SOR 1000 R 0.6 main lobe 0.5
0.4 the m-th side lobes, m =1, 2, 3... X: 0.55 500 0.3 Y: 0.2657 Secrecy outage probability 0.2 X: 0.68 Radius of the lobes of Y: 0.1165 0.1
0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.2 0.4 0.6 0.8 1 Jamming power allocation coefficient φ Jamming power allocation coefficient φ
uj Fig. 3: SOP vs. power allocation coefficient . Parameters Fig. 2: Radius of the main lobe/side lobes of RSOR vs. φ. φ ◦ ◦ Parameters setting: θb = 0 , db = 100m, Nt = 100, and setting: db = 100m, θb = 0 , De = [50m, 100m], Ae = ◦ ◦ Rth = 10bps/Hz. [−15 , 15 ], L = 10, and Rth = 10bps/Hz. estimation as well as nonnegligible multi-cell interference j legitimate user’s direction should be avoided. This conclusion from Alices in adjacent cells. Denote θi,b as the angular coincides with the concept we followed for the design of direction of Bob in Cell i seen from Alice in Cell j, and Cell Algorithm 3. 0 as the objective cell where Bob 0 exists. Some major effects Moreover, as φ increases, the radius of the second side lobe of imperfect CSI and multi-cell interference on the SOR of is reduced first and then increases after a certain value, e.g., Bob 0 are described as follows: φ ≈ 0.7, indicating an optimal jamming power allocation in • Due to pilot contamination from Bobs i, ∀i 6=0, the SOR terms of minimizing the SOP in this direction. The side lobes 0 of Bob 0 will be enlarged in the directions of θi,b. with index m ≥ 3 can be completely eliminated with proper • Multi-cell interference to Bob 0 will isotropically enlarge jamming. The results show that we can design jamming based his SOR. On the other hand, Eves in Cell 0 are equiva- on the partial information of Rsus. For example, in Fig. 2, if lently jammed by the multi-cell interference from Alice j j we know that Eves are located in the direction ranges of side in cell j, ∀j 6=0, especially in the directions of θ0,b and lobes with indices larger than 3, then, allocating φ = 0.2 is j θj,b. Thereby, the SOR in these directions can be shrunk enough to secure the communication and the remaining power and the shape can be non-continuous in these regions. can be allocated to data transmission. A general analytical description of the SOR in the multi- Next, we depict the SOP vs. φ in Fig. 3, with randomly ◦ ◦ cell network is challenging, since it is determined not only by generated θe and de within the range Ae = [−15 , 15 ] and the network topology, but also by the locations of all pilot- De = [50m, 100m], respectively. Note that the smoothless of contaminating users in adjacent cells. Moreover, the compli- the curves is not due to the lack of simulation trials, but caused cated shape of the SOR makes difficulties in calculating and by the fact that Pout is a piecewise function, as shown in (7) minimizing the corresponding area. Nevertheless, in practice, and (30). In addition, the SOP rapidly increases to 1 when pilot scheduling and reuse schemes can be utilized to alleviate φ exceeds φmax in (17). From Fig. 3, we can see that 1) these adverse effects which are caused by pilot contamination, the SOP with optimal φ is much smaller than that without e.g., [34], [35]. jamming, i.e., φ = 0, as anticipated in Proposition 3; 2) the SOP decreases as Nt increases since larger Nt results in higher VII. SIMULATION RESULTS received power at Bob and less leakage to Eve. Moreover, the
Simulation results are shown in this section. We set d0 = optimal φ increases with Nt because with larger Nt, allocating −5 0.5, α = 3, Ptot = 1W, N0 = 10 mW and for simplicity, more power to data transmission is not efficient in increasing assume strong LOS environment such that Ke;b → 1. In the achievable rate of Bob because of the logarithmic slope this parameter setting, the receive SNR is 20dB when the of the rate function; and 3) the SOP is further substantially transmitter-receiver distance is 100m. reduced with directional jamming. At first, using (23) and (24), Fig. 2 provides a description of Fig. 4 shows the optimal jamming power coefficient φopt uj RSOR(φ) in terms of the radius of the main lobe/side lobes. as a function of the normalized crosstalk se;b, for uniform It is shown that the radius of the main lobe is monotonously jamming. We compare the derived φopt in (39) with Monte increasing with φ, indicating that reducing the signal power to- Carlo simulations and show a good match between them. wards θb enlarges the SOP in this direction. Clearly, allocating From Fig. 4, we first observe that each curve is divided into opt additional jamming power to the direction of θb will further two parts, respectively representing that φ0 or φg dominates enlarge this radius, which suggests that jamming directly in the the optimal result in (39). The division is emphasized using 10 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS
) 1 1
opt Monte Carlo
φ Without jamming 0.9 Analytical 0.9 Uniform jamming Directional jamming (Algo. 1) 0.8 opt 0.8 Directional jamming (Algo. 3) φ0 dominates φg dominates 0.7 0.7
0.6 0.6 Nt = 100 0.5 db = 150m 0.5 Nt = 100 d = 100m 0.4 b 0.4 Secrecy outage probability 0.3 0.3 Nt = 50 db = 100m 0.2 0.2
0.1 0.1
Optimal jamming power allocation0 coefficient ( 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 50 60 70 80 90 100 110 120 130 140 150 Normalized crosstalk between Eve and Bob (se;b) db (m)
opt Fig. 4: Optimal jamming power coefficient φ vs. se;b under Fig. 6: Secrecy outage probability vs. db, results are shown Corollary 3 with De = [50m, 350m] and Rth =5 bps/Hz. for no jamming and uniform/directional jamming algorithms. ◦ Parameters setting: Nt = 100, Rth = 10bps/Hz, θb = 0 , ◦ ◦ L = 10, Ae = [−30 , 30 ], and De = [50m, 200m]. 8000 Without jamming
) 7000 2 Uniform jamming m Directional jamming (Algo. 2) uniform jamming, the optimal is found via one dimensional 6000 Directional jamming (Algo. 3) φ linear search by minimizing the area described in (36). For all 5000 curves, the SOR enlarges with increasing db, since larger db results in weaker signal power received at Bob. With uniform 4000 jamming, the area of the SOR can be reduced approximately 3000 by half compared with the non-jamming case. This consider- able reduction of SOR area together with the least implemen- 2000 tation complexity make uniform jamming still a good option
Area of1000 the secrecy outage region ( for practical system design. Using the directional jamming with Algorithm 2, the SOR area can be further reduced, but it 0 50 100 150 induces the highest complexity among all schemes. At last, we db (m) note that for the directional jamming, Algorithm 3 achieves slightly larger area of SOR than Algorithm 2, where the Fig. 5: Area of the secrecy outage region vs. db, results are shown for variant jamming algorithms. Parameters setting: difference becomes smaller especially for small db. However, ◦ the implementation complexity can be greatly reduced by Nt = 50, Rth =5, and θb =0 . Algorithm 3, which makes it being a reasonable choice that strikes a compromise between complexity and performance. It can also be confirmed from Fig. 5 that even without any a vertical dash line for the case with db = 150m. When information of R , directional jamming can still be utilized db = 100m, in the φ0-dominating region, the curves with sus to enhance communication secrecy. Nt = 50 and 100m coincide with each other since Nt does not opt opt affect φ0 in (41). In the φg -dominating region, φ increases We depict the SOP vs. db in Fig. 6 for a particular example ◦ ◦ with Nt with given se;b. The region-division in Fig. 4 can be scenario, where Rsus is defined by Ae = [−30 , 30 ] and explained as follows: in the left region, se;b is small enough De = [50m, 200m]. Four schemes are compared, i.e., without so the channels between Eves and Bob can be considered as jamming, uniform jamming, and directional jamming schemes asymptotically orthogonal, i.e., he ∈ null(hb). In this case, Algorithm 1 and Algorithm 3. Correspondingly, we also depict more signal power is leaked to Eve with larger se;b, hence we the SORs achieved by each schemes in Fig. 7 to help us better need more jamming power allocated in null(hb) to degrade understanding the relation between the SOR and SOP. We note opt Eve’s channel. However, in the φg -dominating region, the that in Fig. 6, the smoothless of the curves is caused by the value of se;b is large, which indicates that channels from Alice area calculation of the intersection between two complicated- to Eves and Bob could be highly aligned, i.e., he ∈/ null(hb). shaped regions, not by the lacking of simulation trials. In this case, jamming in null(hb) is not efficient to degrade As shown in Fig. 6, all jamming schemes can achieve Eve’s channel and the jamming can be a waste of transmit lower SOP compared to that without jamming. In the small-db power. Thus, the optimal jamming power starts decreasing region, directional jamming schemes outperform uniform jam- with se;b. ming. However, the performance improvement from uniform In Fig. 5, for the case that without information of Rsus, we jamming to directional jamming Algorithm 1 is small because compare the area of SOR achieved by different algorithms. For Algorithm 1 directionally allocates jamming power towards 11
(a) Without jamming (b) Uniform jamming 1 90 200 90 200 120 60 120 60 0.9 y = s(x) 100 100 150 30 150 30 y = u 0.8 180 0 180 0 main lobe 0.7 210 330 210 330
0.6 XAi 240 300 240 300 270 270 ) x
( 0.5 s 0.4 (c) Directional jamming (Algo. 1) (d) Directional jamming (Algo. 3) β∗ CP0(u) 90 200 90 200 Ai 120 60 120 60 0.3 CP1,1(u) 150 100 30 150 100 30 0.2 CP1,2(u) 180 0 180 0 0.1 PV1
210 330 210 330 0 0 0.2 0.4 0.6 0.8 1 240 300 240 300 270 270 x
Fig. 8: Description of s(x), the cross points and peak values. Fig. 7: Description of the corresponding SORs for the 4 schemes in Fig. 6, when d = 150m. b the discussions to multiuser and multi-cell scenarios where future challenges are also described. In conclusion, we claim that uniform jamming still helps the communication secrecy Rsus to suppress the SOR side lobes (as shown in Fig. 7 (c)), whereas when Bob is close to Alice, most side lobes in massive MIMO systems, and the proposed directional are already very small with uniform jamming. Moreover, in jamming outperforms conventional uniform jamming schemes. the small-db region, directional jamming Algorithm 3 achieves the lowest SOP for the reason that it focuses on only two APPENDIX A dominating SOR side lobes (as shown in Fig. 7 (d)) hence the PROOF OF LEMMA 1 jamming power can be used more efficiently in Algorithm 3 Proof of 1): We can rewrite si;j (θi,θj ) in (6) as a function than Algorithm 1, where jamming is uniformly performed to of ∆i;j as follows all directions within Rsus. In this example, the two dominating si;j (θi,θj )= Ki;j s (∆i;j ) . (52) side lobes are covered by Rsus, thereby they contribute more in the SOP calculation compared with the other side lobes. By applying [31, (14)] to ti;j in (6), s(x) in (52) can be However, in a different scenario where Rsus does not cover represented as the two dominating side lobes, it cannot be concluded that 1, x =0 Algorithm 3 always outperforms Algorithm 1. 2 s (x)= 1 sin (Ntπdx) (53) At last, we note that as increases, the performance 2 2 , x 6=0 db ( Nt sin (πdx) of Algorithm 3 degrades and Algorithm 1 outperforms the which is a sinc-like function, has one main lobe and side lobes others. The reason is, when d is large, all SOR side lobes b with decreasing amplitudes. correspondingly become large as shown in Fig. 7 (d). In this Proof of 2): In order to describe the CDF of s , we first case, besides the two dominating side lobes, the impact from i;j characterize s(x) in (53) by some cross points and peak values, the other side lobes cannot be simply ignored as that has been shown in Fig. 8 and defined in the following. done in Algorithm 3. In conclusion, in practice, appropriate Definition 4: The cross points and peak values are defined jamming scheme should be determined according to both the as available information of Rsus and the location information of • Peak Values: The peak value of the m-th side lobe of Bob such as db. s(x) can be approximated by VIII. CONCLUSIONS 1 1 1 PVm ≈ ≈ (54) 2 m+ 1 2 In this paper, we first define and analytically describe the Nt 2 2 2 1 sin π N π m + 2 SOR for secure communication in massive MIMO Rician t channels, and derive expressions of the SOP. We then de- which is obtained by noting that sin( x) ≈ x when x is termine a jamming-beneficial range, indicating that uniform small, and it becomes asymptotically exact as Nt is large. jamming is useful in reducing the SOP when the distance PV0 =1 corresponds to the main lobe. from Eve to Alice is less than a threshold. Optimal jam- • Cross Points: When u < PVM , CPm,i(u),i = 1, 2 ming and signal power allocation is investigated for uniform denotes the i-th cross point between y = u and y = s(x) jamming, furthermore, for both conditions with and without in side lobe m (m α calculating the probability that ∆i;j falls within the discrete then z < a2 holds for any z. In this case, (58) can be intervals determined by CPm,i(u) and CP0(u). On the other equivalently rewritten as hand, recalling that ∆i;j , | sin θi − sin θj | and θi follows α ˜ (a1 − a2)z uniform distribution, the CDF of ∆i;j can be written as Pjam > α . (60) a2 − z dh(x) F∆(z) ˜ Noting that dx < 0, it is clear from (60) that Pjam > 0, 1 −1 thus (59) is a sufficient condition to satisfy 1). On the other , min sin (min (1,z + sin θ )) ,θ dα j max hand, as t ≤ max , a sufficient condition that satisfies 2) is θmax − θmin 0 a2 + α max −1 d m m+1 where Am = [ N d , N d ] is the physical angle range of the 1) Assuming that the angle ranges occupied by every side t t dj m-th side lobe. To obtain (66), we use α = 2, and (67) is lobe (and the main lobe) of RSOR(p¯) are approximately π obtained using (65). From (67) to (68), we use the variable the same (which is M+1 assuming that there are in total substitution η = Ntπdθe, then (69) is obtained by simple one main lobe and M side lobes within the angle range π π integral calculation of elementary functions. 2 , 2 ), an upper bound of the integral in (47) can be approximated by APPENDIX D M π 2 PROOF OF COROLLARY 3 Area Rdj (p¯) ≈ d¯dj(p¯,θ ) UB SOR 2(M + 1) e m When , , in (30) is determined m=0 θel = θe ∀l = 1, ..., L Pout X (73) only by de such that where the area of every side lobe is upper bounded by L the area of its enclosing sector. dmax 2z (70) 2) We assume that ¯s(θ )H v = 1 and ¯s(θ )H v = Pout =1 − 2 2 dz m m n m d − d ¯s(θm) max[d0(φ,θe),dmin] max min ! 0, ∀n 6= m. In practice, by defining v = , Z m ||¯s(θm)|| where this assumption is easy to realize with large Nt, where 1 asymptotic orthogonality holds. Now, according to (44), ˜ ˜ ˜ α ¯dj d0(φ, θe)= h (1 − φ)Ptot + φPtot se;b(θe) − φPtot de (p¯,θm) in (73) can be written as 1 1 + , hn α o i [g(φ)] (71) ¯dj ˜ α de (p¯,θm)= am − Pm (74) is the distance threshold. Given θe, any Eve with a distance φ P˜ N Rth s θ ¯ to Alice smaller than d0(φ, θe) can cause secrecy outage. (1− ) tot t2 e;b( m) ˜ Pm where am = −α and Pm = 1+(1−φ)P˜ d N −2Rth N0 Noting this, (70) calculates the overall outage probability tot b t is the -th element of p, with ¯ being the jamming by assuming that d follows uniform distribution. Clearly, if m ˜ Pm e power allocated on the m-th side lobe. min d0(φ, θe) < dmin, the integral range in (70) becomes φ From (73) and (74), and given fixed φ, we rewrite the original [d , d ] hence a zero SOP can be achieved; otherwise min max area minimization problem as if min d0(φ, θe) > dmax, a definite outage occurs with proba- φ M 2 bility 1.5 Rewrite g(φ) in (71) as α min am − P˜m (75) , m=1 g(φ) se;b(θe)g1(φ)+(1 − se;b(θe))g2(φ) (72) XM , ˜ , ˜ where g1(φ) h((1 − φ)Ptot), g2(φ) −φPtot and h(·) s.t. P¯m = φPtot, 0 ≤ P˜m ≤ am. (76) is defined in (56). Clearly, minimizing (70) is equivalent to m=0 X minimizing (72). Since g1(φ) is concave and g2(φ) is linear, By checking the Hessian matrix, it is easy to show that the ∂g(φ) g(φ) is concave. Hence, letting ∂φ = 0, the optimal φ objective function in (75) is concave. To minimize a concave max opt satisfying φ < φ (as in (17)) is given by φg in (40). function, clearly, the optimal solution can be found only opt opt opt Note that if d0(φg ,θe) < dmin, setting φ = φg is on the boundaries of the domain defined by (76). Recalling not the only choice since the solution of d0(φ, θe)= dmin (if that the area of the side lobes decreases rapidly with larger exists), denoted as φ0, also achieves zero SOP. Therefore, we lobe index, and being aware that jamming should be avoided opt need to check the value of φ0 and compare it with φg to within the main lobe to prevent degrading Bob’s channel, we determine the final optimal jamming power allocation. Note further simplify the problem by checking the boundary of the that in (72), the value of g1(φ) is usually much less than domain as described in (48). According to these discussions, g2(φ). Hence, when se;b(θe) is not so large, g(φ) can be well Algorithm 3 is obtained. approximated by a linear function as g(φ) ≈ se;b(θe)h(P˜tot)− (1 − se;b(θe))P˜totφ. Using this approximation, we get φ0 = s θ h P˜ dα REFERENCES e;b( e) ( tot)− min . (1−se;b(θe))P˜tot If φ0 ∈/ [0, 1], it is not a feasible solution in practice. If [1] F. Rusek, D. Persson, B. K. Lau, E. G. Larsson, T. L. Marzetta, opt O. Edfors, and F. Tufvesson, “Scaling up MIMO: Opportunities and φ0 ∈ [0, 1], both φ0 and φg are able to achieve zero SOP. In opt challenges with very large arrays,” IEEE Signal Process. Mag., vol. 30, this case, we choose the smaller one between φg and φ0 to no. 1, pp. 40–46, Jan. 2013. opt opt save jamming power such that φ = min{φg , φ0}. [2] E. G. Larsson, F. Tufvesson, O. Edfors, and T. L. Marzetta, “Massive MIMO for next generation wireless systems,” IEEE Commun. Mag., vol. 52, no. 2, pp. 186–195, Feb. 2014. APPENDIX E [3] T. L. 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His research interests include MIMO wireless communications, “Principles of physical layer security in multiuser wireless networks: A multiuser transmission, MIMO channel modeling, massive MIMO systems survey,” IEEE Comm. Surveys and Tutorials, vol. 16, no. 3, pp. 1550– and physical layer security. 1573, Aug. 2014. [20] G. Geraci, M. Egan, J. Yuan, A. Razi, and I. B. Collings, “Secrecy sum- rates for multi-user MIMO regularized channel inversion precoding,” IEEE Trans. Commun., vol. 60, no. 11, pp. 3472–3482, Nov. 2012. [21] G. Geraci, R. Couillet, J. Yuan, M. Debbah, and I. B. Collings, “Large system analysis of linear precoding in MISO broadcast channels with Jemin Lee (S’06-M’11) is a Temasek Research confidential messages,” IEEE J. Sel. Areas Commun., vol. 31, no. 9, pp. Fellow at the Singapore University of Technology 1660–1671, Sep. 2013. and Design (SUTD), Singapore. She received the [22] G. Geraci, S. Singh, J. G. Andrews, J. Yuan, and I. B. Collings, “Secrecy B.S. (with high honors), M.S., and Ph.D. degrees in rates in broadcast channels with confidential messages and external Electrical and Electronic Engineering from Yonsei eavesdroppers,” IEEE Trans. Wireless Commun., vol. 13, no. 5, pp. University, Seoul, Korea, in 2004, 2007, and 2010, 2931–2943, May 2014. respectively. She was a Postdoctoral Fellow at the [23] J. Zhang, C. Yuen, C.-K. Wen, S. Jin, and X. Gao, “Ergodic secrecy Massachusetts Institute of Technology (MIT), Cam- sum-rate for multiuser downlink transmission via regularized channel bridge, MA from Oct. 2010 to Oct. 2013, and a inversion: Large system analysis,” IEEE Commun. Letters, vol. 18, no. Visiting Ph.D. Student at the University of Texas 9, pp. 1627–1630, Sep. 2014. at Austin, Austin, TX from Dec. 2008 to Dec. [24] Q. Xiong, Y.-C. Liang, K. H. Li, and Y. Gong, “An energy-ratio- 2009. Her current research interests include physical layer security, wireless based approach for detecting pilot spoofing attack in multiple-antenna security, heterogeneous networks, cognitive radio networks, and cooperative systems,” IEEE Trans. Inf. Forensics Security, vol. 10, no. 5, pp. 932– communications. 940, May 2015. Dr. Lee is currently an Editor for the IEEE TRANSACTIONS ON WIRELESS [25] D. Kapetanovi´c, G. Zhang, K.-K. Wong, and B. Ottersten, “Detection of COMMUNICATIONS and the IEEE COMMUNICATIONS LETTERS, and served pilot contamination attack using random training and massive MIMO,” as a Guest Editor of the Special Issue on Heterogeneous and Small Cell in Proc. IEEE Int. Symp. Personal Indoor Mobile Radion Commun., Networks for the ELSEVIER PHYSICAL COMMUNICATION in 2014. She London, U. K. Sep. 2013, pp. 13–18. also served as a Co-Chair of the IEEE 2013 Globecom Workshop on [26] X. Zhou, B. Maham, and A. Hjørungnes, “Pilot contamination for active Heterogeneous and Small Cell Networks, and Technical Program Committee eavesdropping,” IEEE Trans. Wireless Commun., vol. 11, no. 3, pp. 903– Member for numerous IEEE conferences. She is currently a reviewer for 907, Mar. 2012. several IEEE journals and has been recognized as an Exemplary Reviewer [27] S. Goel and R. Negi, “Guaranteeing secrecy using artificial noise,” IEEE of IEEECOMMUNICATIONS LETTERS and IEEE WIRELESS COMMUNICA- Trans. Wireless Commun., vol. 7, no. 6, pp. 2180–2189, Jun. 2008. TION LETTERS for recent several years. She received the IEEE ComSoc Asia- [28] J. Zhu, R. Schober, and V. K. Bhargava, “Secure transmission in multi- Pacific Outstanding Young Researcher Award in 2014, the Temasek Research cell massive MIMO systems,” IEEE Trans. Wireless Commun., vol. 13, Fellowship in 2013, the Chun-Gang Outstanding Research Award in 2011, no. 9, pp. 4766–4781, Sep. 2014. and the IEEE WCSP Best Paper Award in 2014. [29] J. Zhu, R. Schober, V. K. Bhargava, “Secure downlink transmission in massive MIMO system with zero-forcing precoding,” in Proc. European Wireless Conf., May 2014, pp. 1–6. [30] Q. Zhang, Z. Lu, S. Jin, K.-K. Wong, H. Zhu, and M. Matthaiou, “Power scaling of massive MIMO systems with arbitrary-rank channel means 15 Fanggang Wang (S’10-M’11) received the B.Eng. degree in 2005 and the Ph.D. degree in 2010 from the School of Information and Communica- tion Engineering at Beijing University of Posts and Telecommunications, Beijing, China. From 2008 to 2010, he worked as a Visiting Scholar in Electrical Engineering Department, Columbia University, New York City, New York, USA. He was a Postdoctoral Fellow in Institute of Network Coding, the Chinese University of Hong Kong, Hong Kong SAR, China, from 2010 to 2012. He joined the State Key Lab of Rail Traffic Control and Safety, School of Electronic and Information Engineering, Beijing Jiaotong University, in 2010, where he is currently an Associate Professor. His research interests are in wireless communications, signal processing, and information theory. He chaired two workshops on wireless network coding (NRN 2011 and NRN 2012) and served as an Editor in several journals and the Technical Program Committee (TPC) members in several conferences. Tony Q. S. Quek (S’98-M’08-SM’12) received the B.E. and M.E. degrees in Electrical and Electronics Engineering from Tokyo Institute of Technology, Tokyo, Japan, respectively. At Massachusetts Insti- tute of Technology, he earned the Ph.D. in Electrical Engineering and Computer Science. Currently, he is an Assistant Professor with the Information Systems Technology and Design Pillar at Singapore Univer- sity of Technology and Design (SUTD). He is also a Scientist with the Institute for Infocomm Research. His main research interests are the application of mathematical, optimization, and statistical theories to communication, net- working, signal processing, and resource allocation problems. Specific current research topics include heterogeneous networks, green communications, smart grid, wireless security, internet-of-things, big data processing, and cognitive radio. Dr. Quek has been actively involved in organizing and chairing sessions, and has served as a member of the Technical Program Committee as well as symposium chairs in a number of international conferences. He is serving as the technical chair for the PHY & Fundamentals Track for IEEE WCNC in 2015, the Communication Theory Symposium for IEEE ICC in 2015, the PHY & Fundamentals Track for IEEE EuCNC in 2015, and the Communication and Control Theory Symposium for IEEE ICCC in 2015. He is currently an Editor for the IEEE TRANSACTIONS ON COMMUNICATIONS,the IEEEWIRELESS COMMUNICATIONS LETTERS, and an Executive Editorial Committee Mem- ber for the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS. He was Guest Editor for the IEEE SIGNAL PROCESSING MAGAZINE (Special Issue on Signal Processing for the 5G Revolution) in 2014, and the IEEE WIRELESS COMMUNICATIONS MAGAZINE (Special Issue on Heterogeneous Cloud Radio Access Networks) in 2015. Dr. Quek was honored with the 2008 Philip Yeo Prize for Outstanding Achievement in Research, the IEEE Globecom 2010 Best Paper Award, the CAS Fellowship for Young International Scientists in 2011, the 2012 IEEE William R. Bennett Prize, the IEEE SPAWC 2013 Best Student Paper Award, and the IEEE WCSP 2014 Best Paper Award. 0. With θb = 0, we can rewrite x (defined as x = L | sin θ − sin θ |) in terms of θ as dmax 2z e b e Pout =1 − Fse;b (t) × 2 dz . (57) d2 − d x = sin θe ≤ θe. (64) ( Zdmin max min ) Note that the side lobes that are close to the main lobe are more Since Fse;b (·) is an increasing function, Pout decreases with important in contributing to the area of the SOR, as a result, the t in the domain of Fse;b (·). Therefore, if the following condi- tions are satisfied, we can conclude that jamming is beneficial. value of θe that should be concerned is very small. Therefore, the upper bound in (64) can be very tight. Combining (63) ˜ 1) There exists a positive value of Pjam, which holds t − and (64) we have t0 > 0 for any z ∈ De; and max Ke;b 2 2) z ∈ De such that t0